Problem 26 - Algebra 1

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 358=39 c-17 $$

Step-by-Step Solution

Answer

The solution for c is approximately 9.62 when rounded to the nearest hundredth.

Step 1: Isolate the variable term

First, let's move the constant term from the right side of the equation to the left side. To do this, add 17 to both sides of the equation.\[ 358 + 17 = 39c \]This simplifies to:\[ 375 = 39c \]

Step 2: Solve for the variable

Next, divide both sides of the equation by 39 to get c by itself.\[ c = \frac{375}{39} \]

Step 3: Round to the nearest hundredth

When you compute the division, you get c approximately equal to 9.62 when rounded to the nearest hundredth.

Step 4: Check the solution

Let's check the solution by substituting c = 9.62 into the original equation:\[ 358 = 39 \cdot 9.62 - 17 \]Rounding the result of \(39 \cdot 9.62\) to the nearest hundredth and subtracting 17 should yield 358 if our solution is correct.

Key Concepts

Isolating the Variable

In any linear equation, the primary goal is to isolate the variable to one side of the equation. This allows us to directly find its value. The process involves moving terms to different sides of the equation by using basic arithmetic operations, such as addition or subtraction.
Let's revisit the given equation:

• Start with: \(358 = 39c - 17\).
• Add 17 to both sides to remove the constant from the right side.
• This results in: \(375 = 39c\).

By performing these operations, we've gathered all terms involving the variable, \(c\), on one side. This crucial step simplifies the equation and prepares it for solving the variable.

Rounding Numbers

Once you solve for the variable, sometimes the result might be a long decimal. This isn't always practical, so rounding is used to simplify the number, making it easier to work with.
After isolating the variable, we found:

• \(c = \frac{375}{39}\).

When you compute this division, you get approximately \(c \approx 9.61538\). When the problem asks us to round to the nearest hundredth, it means we want two decimal places. Look at the third decimal to decide whether to round up or keep it the same. Here:

• Since the third decimal is 5, we round up to 9.62.

Checking Solutions

After computing a solution and rounding it, it's vital to check if it satisfies the original equation. This double-checks our calculations and offers assurance that our answer is correct.
For the rounded value \(c = 9.62\):

• Substitute it back into the original equation: \(358 = 39 \times 9.62 - 17\).
• Calculate the right-hand side to check if it equals 358:\(39 \times 9.62 = 375.18\).
• Next, \(375.18 - 17 = 358.18\).

Since this is close to 358, if we initially didn't round on earlier steps, minor discrepancies might arise due to this rounding. However, this closeness confirms our rounded solution is accurate.

Linear Equations

Linear equations are one of the foundational building blocks in algebra. They describe relationships where one variable is directly proportional to another. The general form for a linear equation is \(ax + b = c\) where \(a\), \(b\), and \(c\) are constants.
In this exercise, the linear equation given was \(358 = 39c - 17\).

• Coefficients and constants like 39 and -17 represent fixed values determining the slope and intercept of the equation.
• Each operation we performed was aimed at simplifying this form to \(c =\) some number.

By understanding and manipulating linear equations, we can solve various real-world problems through this straightforward relationship.